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3.4 Approximation Types 31

training or design points D train within the domain of interest D train D

x
X. x denotes the input variable set x fx d g X, w a parameter
set w w w .
A good measure for the quality of the approximation will depend on
the intended task. However, in most cases accuracy is of importance and is

measured employing a distance function dist f x w Fx (e.g. L2 norm).
The lack of accuracy or lack-of-fit LO F is often defined by the expected
F
error

F
LO F D hdist f x w Fx i D (3.1)
the dist function averaged over the domain of interest D. The approx-

imation problem is to determine the parameter set w , that minimizes
F F D . The ultimate solution depends strongly on F. If it exists, it is
LO
called “best approximation” (see e.g. Davis 1975; Poggio and Girosi 1990;
Friedman 1991).

Summarizing, several main problems in building a learning system can be
distinguished:

(i) encoding the problem in a suitable representation x;

(ii) finding a suitable approximation function F;

(iii) choosing the algorithm to find optimal values for the parameters W;

(iv) the problem of efficiently implementing the algorithm.

The proceeding chapter 4 will present the PSOM approach with respect
to (ii)–(iv). Numerous examples for (i) are presented in the later chapters.
The following section discusses several common methods for (ii)

3.4 Approximation Types

In the following we consider some prominent examples of approximat-
I
ing functions F w x R d IR – for the moment simplified to one-
dimensional outputs.

The classical linear case is

F w x w x (3.2)